Fix a base point \(x_0\, \in \, \Sigma _{n,g}\) . When \(G\)  is a reductive affine algebraic group defined over \(\mathbb {C}\) ,
a homomorphism \(\rho \, :\, \pi _1(\Sigma _{n,g},\, x_0)\, \longrightarrow \, G\)  is called reductive if the
Zariski closure of \(\rho (\pi _1(\Sigma _{n,g},\, x_0))\)  in \(G\)  is a reductive subgroup. We note that \(\rho \)  is
reductive if and only if satisfies the following condition: If \(P\)  is a parabolic subgroup of \(G\)  such that
\(\rho (\pi _1(\Sigma _{n,g},\, x_0))\, \subset \, P\) , then \(\rho (\pi _1(\Sigma _{n,g},\, x_0))\)  is contained in a Levi
factor of \(P\)  (see [1]} and [2]} for parabolic subgroups and their Levi factors).
